Expanding a binomial with a high exponent such as [latex]{\left(x+2y\right)}^{16}[/latex] can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of [latex]{\left(x+y\right)}^{5}[/latex].
The second term is [latex]\left(\begin{array}{c}5\\ 1\end{array}\right){x}^{4}y[/latex]. The third term is [latex]\left(\begin{array}{c}5\\ 2\end{array}\right){x}^{3}{y}^{2}[/latex]. We can generalize this result.
A General Note: The (r+1)th Term of a Binomial Expansion
The [latex]\left(r+1\right)\text{th}[/latex] term of the binomial expansion of [latex]{\left(x+y\right)}^{n}[/latex] is:
How To: Given a binomial, write a specific term without fully expanding.
- Determine the value of [latex]n[/latex] according to the exponent.
- Determine [latex]\left(r+1\right)[/latex].
- Determine [latex]r[/latex].
- Replace [latex]r[/latex] in the formula for the [latex]\left(r+1\right)\text{th}[/latex] term of the binomial expansion.
Example 3: Writing a Given Term of a Binomial Expansion
Find the tenth term of [latex]{\left(x+2y\right)}^{16}[/latex] without fully expanding the binomial.
Solution
Because we are looking for the tenth term, [latex]r+1=10[/latex], we will use [latex]r=9[/latex] in our calculations.
Try It 3
Find the sixth term of [latex]{\left(3x-y\right)}^{9}[/latex] without fully expanding the binomial.